Expand the Logarithmic Expression natural log of x^2 square root of y/z

Problem

ln((x2√(,y))/z)

Solution

1. Apply the quotient rule for logarithms, which states that ln(a/b)=ln(a)−ln(b)

ln((x2√(,y))/z)=ln(x2√(,y))−ln(z)

2. Apply the product rule for logarithms to the first term, which states that ln(a*b)=ln(a)+ln(b)

ln(x2√(,y))−ln(z)=ln(x2)+ln(√(,y))−ln(z)

3. Rewrite the radical using a fractional exponent, where √(,y)=y(1/2)

ln(x2)+ln(y(1/2))−ln(z)

4. Apply the power rule for logarithms, which states that ln(an)=n*ln(a) to both the first and second terms.

2*ln(x)+1/2*ln(y)−ln(z)

Final Answer

ln((x2√(,y))/z)=2*ln(x)+1/2*ln(y)−ln(z)

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